Page 375 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 359

6.6 Incompressible Newtonian Fluid

For an incompressible fluid, A = £>,-,- = 0 at all times. Thus, the constitutive equation for
such a fluid becomes

We see from this equation that

Thus,

Therefore, for an incompressible viscous fluid, the pressure has the meaning of the mean
normal compressive stress. The value of p does not depend explicitly on any kinematic
quantities; its value is indeterminate as far as the fluid's mechanical behavior is concerned. In
other words, since the fluid is incompressible, one can superpose any pressure to the fluid,
without affecting its mechanical behavior. Thus, the pressure in an incompressible fluid is
often known constitutively as the "indeterminate pressure". In any given problem with
prescribed boundary condition(s) for the pressure, the pressure field is determinate.
Since

where v/ are the velocity components, the constitutive equations can be written:

i.e.,